Abnormal subgroups in solvable groups of finite Morley rank
type: Item
instance of: scholarly article
title: Abnormal subgroups in solvable groups of finite Morley rank
zbMATH Open document ID: 0984.20022
published in: Journal of Algebra
publication date: 2002-02-06T00:00:00Z
review text: The author begins a systematic study of the theory of formations in connected soluble groups of finite Morley rank. Recall that a subgroup \(H\) of a group \(G\) is abnormal if \(g\in\langle H,H^g\rangle\) for any \(g\in G\); it is a Carter subgroup if it is nilpotent and self-normalizing. The author proves that in a connected soluble group of finite Morley rank minimal abnormal subgroups exist and are conjugate; they are precisely the Carter subgroups (Théorème 1.1). Moreover, the following are equivalent: (1) \(H\) is abnormal. (2) \(H\) contains a Carter subgroup. (3) \(H\) is definable, and any \(g\in G\) is contained in the smallest definable supergroup of \(H\cup H^g\). (4) \(H\) is definable, connected, and there is a sequence \(G=H_0>H_1>\cdots>H_n=H\) such that \(H_{i+1}\) is a proper connected definable non-normal subgroup of \(H_i\) for all \(i<n\). (Théorème 1.2.) This extends, in the context of groups of finite Morley rank, earlier work of \textit{F. Wagner} [Arch. Math. Logic 33, No. 1, 23-34 (1994; Zbl 0813.03022)], who proved existence and conjugacy for Carter subgroups in the larger class of ``very'' connected small stable groups. In a final section, the author develops a theory of connected saturated formations for soluble groups of finite Morley rank; for such a formation, covering subgroups exist and are conjugate (Théorème 8.5). By his previous results, the class of connected nilpotent groups of finite Morley rank is a connected saturated formation, whose covering subgroups are precisely the Carter subgroups. It is to be expected that these results will have many applications in the study of soluble groups of finite Morley rank. A first application is by \textit{E. Jaligot} [J. Algebra 240, No. 2, 413-444 (2001; Zbl 0994.20028)]. In the introduction the author claims that B. Poizat has constructed a bad field. This is not quite true: \textit{B. Poizat} [J. Symb. Log. 64, No. 3, 1339-1355 (1999; Zbl 0938.03058)], and \textit{J. T. Baldwin} and \t…
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F17
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 03C45
Mathematics Subject Classification ID: 03C60
zbMATH DE Number: 1496007
zbMATH Keywords: soluble groups
zbMATH Keywords: abnormal subgroups
zbMATH Keywords: Carter subgroups
zbMATH Keywords: saturated formations
zbMATH Keywords: groups finite Morley rank
zbMATH Keywords: definable subgroups
zbMATH Keywords: stable groups
zbMATH Keywords: covering subgroups
author: Frécon, Olivier
MaRDI profile type: Publication
full work available at URL: jabr.2000.8302
OpenAlex ID: W1972791737
cites work: A Hall Theorem for ω-Stable Groups
cites work: Q4317873
cites work: On the Schur-Zassenhaus theorem for groups of finite Morley rank
cites work: Schur-Zassenhaus theorem revisited
cites work: Periodic locally soluble groups with the minimal condition on centralizers
cites work: Nilpotent self-normalizing subgroups of soluble groups
cites work: Saturated formations and Sylow structure in locally finite groups
cites work: Q5662113
cites work: Zur Theorie der endlichen auflösbaren Gruppen
cites work: The Engel elements of a soluble group
cites work: Groups of finite Morley rank of even type with a weakly embedded subgroup
cites work: Strong finiteness conditions in locally finite groups
cites work: On Solvable Groups of Finite Morley Rank
cites work: Poly-separated and ω-stable nilpotent groups
cites work: Generalized Fitting subgroup of a group of finite Morley rank
cites work: Q3961021
cites work: Nilpotent complements and Carter subgroups in stable \({\mathfrak R}\)
cites work: Q4355426
cites work: Группы и кольца, теория которых категорична
cites work: Some model theory of simple algebraic groups over algebraically closed fields
cites work: On groups with nilpotent derived groups
DOI: 10.1006/JABR.2000.8302
Recommended article: Sous-groupes de Carter dans les groupes de rang de Morley fini
Recommended article: SIMPLE GROUPS OF MORLEY RANK 5 ARE BAD
Recommended article: Q3602603
Recommended article: Bad groups of finite Morley rank
Recommended article: Generix never gives up
Recommended article: The existence of Carter subgroups in groups of finite Morley rank
Recommended article: Sylow theory for p = 0 in solvable groups of finite Morley rank
Recommended article: Groups of Morley rank 4
Recommended article: Q3770506
Recommended article: Splitting in solvable groups of finite Morley rank
description: scientific article; zbMATH DE number 1496007